Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
pi*(2*cos(2x-1))/((tg(pi*x)^2)*(cos(pi*x)^2))
¿Cómo vas a descomponer esta pi*(2*cos(2x-1))/((tg(pi*x)^2)*(cos(pi*x)^2)) expresión en fracciones?
Solución
Ha introducido
[src]
pi*2*cos(2*x - 1) --------------------- 2 2 tan (pi*x)*cos (pi*x)
$$\frac{\pi 2 \cos{\left(2 x - 1 \right)}}{\cos^{2}{\left(\pi x \right)} \tan^{2}{\left(\pi x \right)}}$$
(pi*(2*cos(2*x - 1)))/((tan(pi*x)^2*cos(pi*x)^2))
Simplificación general
[src]
4*pi*cos(-1 + 2*x) ------------------ 1 - cos(2*pi*x)
$$\frac{4 \pi \cos{\left(2 x - 1 \right)}}{1 - \cos{\left(2 \pi x \right)}}$$
4*pi*cos(-1 + 2*x)/(1 - cos(2*pi*x))
Respuesta numérica
[src]
6.28318530717959*cos(2*x - 1)/(cos(pi*x)^2*tan(pi*x)^2)
6.28318530717959*cos(2*x - 1)/(cos(pi*x)^2*tan(pi*x)^2)
Denominador común
[src]
2*pi*cos(-1 + 2*x) --------------------- 2 2 cos (pi*x)*tan (pi*x)
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\cos^{2}{\left(\pi x \right)} \tan^{2}{\left(\pi x \right)}}$$
2*pi*cos(-1 + 2*x)/(cos(pi*x)^2*tan(pi*x)^2)
Combinatoria
[src]
2*pi*cos(-1 + 2*x) --------------------- 2 2 cos (pi*x)*tan (pi*x)
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\cos^{2}{\left(\pi x \right)} \tan^{2}{\left(\pi x \right)}}$$
2*pi*cos(-1 + 2*x)/(cos(pi*x)^2*tan(pi*x)^2)
Potencias
[src]
2*pi*cos(-1 + 2*x) --------------------- 2 2 cos (pi*x)*tan (pi*x)
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\cos^{2}{\left(\pi x \right)} \tan^{2}{\left(\pi x \right)}}$$
2
/ pi*I*x -pi*I*x\ / I*(1 - 2*x) I*(-1 + 2*x)\
-pi*\e + e / *\e + e /
---------------------------------------------------------
2
/ pi*I*x -pi*I*x\ 2
|e e | / pi*I*x -pi*I*x\
|------- + --------| *\- e + e /
\ 2 2 /
$$- \frac{\pi \left(e^{i \left(1 - 2 x\right)} + e^{i \left(2 x - 1\right)}\right) \left(e^{i \pi x} + e^{- i \pi x}\right)^{2}}{\left(- e^{i \pi x} + e^{- i \pi x}\right)^{2} \left(\frac{e^{i \pi x}}{2} + \frac{e^{- i \pi x}}{2}\right)^{2}}$$
-pi*(exp(pi*i*x) + exp(-pi*i*x))^2*(exp(i*(1 - 2*x)) + exp(i*(-1 + 2*x)))/((exp(pi*i*x)/2 + exp(-pi*i*x)/2)^2*(-exp(pi*i*x) + exp(-pi*i*x))^2)
Unión de expresiones racionales
[src]
2*pi*cos(-1 + 2*x) --------------------- 2 2 cos (pi*x)*tan (pi*x)
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\cos^{2}{\left(\pi x \right)} \tan^{2}{\left(\pi x \right)}}$$
2*pi*cos(-1 + 2*x)/(cos(pi*x)^2*tan(pi*x)^2)
Denominador racional
[src]
2*pi*cos(-1 + 2*x) --------------------- 2 2 cos (pi*x)*tan (pi*x)
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\cos^{2}{\left(\pi x \right)} \tan^{2}{\left(\pi x \right)}}$$
2*pi*cos(-1 + 2*x)/(cos(pi*x)^2*tan(pi*x)^2)
Compilar la expresión
[src]
2*pi*cos(2*x - 1) --------------------- 2 2 cos (pi*x)*tan (pi*x)
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\cos^{2}{\left(\pi x \right)} \tan^{2}{\left(\pi x \right)}}$$
2*pi*cos(2*x - 1)/(cos(pi*x)^2*tan(pi*x)^2)
Parte trigonométrica
[src]
2 2 / 2 \
pi*(1 + cos(pi*x)) *csc (pi*x)*\1 - tan (-1/2 + x)/
---------------------------------------------------
/ 2 \ 4/pi*x\
2*\1 + tan (-1/2 + x)/*cos |----|
\ 2 /
$$\frac{\pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right) \left(\cos{\left(\pi x \right)} + 1\right)^{2} \csc^{2}{\left(\pi x \right)}}{2 \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \cos^{4}{\left(\frac{\pi x}{2} \right)}}$$
/ 4 \
2 | 4*sin (-1/2 + x)|
pi*sin (2*pi*x)*|1 - ----------------|
| 2 |
\ sin (-1 + 2*x) /
------------------------------------------------------------------
2
/ 4/pi*x\\
| 4*sin |----|| / 4 \
| \ 2 /| | 4*sin (-1/2 + x)| 4/pi*x\ 4
2*|1 - ------------| *|1 + ----------------|*cos |----|*sin (pi*x)
| 2 | | 2 | \ 2 /
\ sin (pi*x) / \ sin (-1 + 2*x) /
$$\frac{\pi \left(- \frac{4 \sin^{4}{\left(x - \frac{1}{2} \right)}}{\sin^{2}{\left(2 x - 1 \right)}} + 1\right) \sin^{2}{\left(2 \pi x \right)}}{2 \left(- \frac{4 \sin^{4}{\left(\frac{\pi x}{2} \right)}}{\sin^{2}{\left(\pi x \right)}} + 1\right)^{2} \left(\frac{4 \sin^{4}{\left(x - \frac{1}{2} \right)}}{\sin^{2}{\left(2 x - 1 \right)}} + 1\right) \sin^{4}{\left(\pi x \right)} \cos^{4}{\left(\frac{\pi x}{2} \right)}}$$
/ pi \
4*pi*sin|-1 + -- + 2*x|
\ 2 /
-----------------------
/pi \
1 - sin|-- + 2*pi*x|
\2 /
$$\frac{4 \pi \sin{\left(2 x - 1 + \frac{\pi}{2} \right)}}{1 - \sin{\left(2 \pi x + \frac{\pi}{2} \right)}}$$
/ 2 \
pi*\1 - tan (-1/2 + x)/
--------------------------------------------
/ 2 \ 4/pi*x\ 2/pi*x\
2*\1 + tan (-1/2 + x)/*cos |----|*tan |----|
\ 2 / \ 2 /
$$\frac{\pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right)}{2 \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \cos^{4}{\left(\frac{\pi x}{2} \right)} \tan^{2}{\left(\frac{\pi x}{2} \right)}}$$
/ 4 \
2 | 4*sin (-1/2 + x)|
pi*sin (2*pi*x)*|1 - ----------------|
| 2 |
\ sin (-1 + 2*x) /
-----------------------------------------------------------------------
2
/ 4/pi*x\\
| 4*sin |----|| / 4 \
| \ 2 /| | 4*sin (-1/2 + x)| 4 4/pi pi*x\
2*|1 - ------------| *|1 + ----------------|*sin (pi*x)*sin |-- + ----|
| 2 | | 2 | \2 2 /
\ sin (pi*x) / \ sin (-1 + 2*x) /
$$\frac{\pi \left(- \frac{4 \sin^{4}{\left(x - \frac{1}{2} \right)}}{\sin^{2}{\left(2 x - 1 \right)}} + 1\right) \sin^{2}{\left(2 \pi x \right)}}{2 \left(- \frac{4 \sin^{4}{\left(\frac{\pi x}{2} \right)}}{\sin^{2}{\left(\pi x \right)}} + 1\right)^{2} \left(\frac{4 \sin^{4}{\left(x - \frac{1}{2} \right)}}{\sin^{2}{\left(2 x - 1 \right)}} + 1\right) \sin^{4}{\left(\pi x \right)} \sin^{4}{\left(\frac{\pi x}{2} + \frac{\pi}{2} \right)}}$$
2 2 4/pi*x\ / 1 \
2*pi*cos (pi*x)*csc (pi*x)*csc |----|*|-1 + --------------|
\ 2 / | 2 |
\ tan (-1/2 + x)/
-----------------------------------------------------------
2
/ 1 \ 2
|-1 + ----------| *csc (-1/2 + x)
| 2/pi*x\|
| tan |----||
\ \ 2 //
$$\frac{2 \pi \left(-1 + \frac{1}{\tan^{2}{\left(x - \frac{1}{2} \right)}}\right) \cos^{2}{\left(\pi x \right)} \csc^{4}{\left(\frac{\pi x}{2} \right)} \csc^{2}{\left(\pi x \right)}}{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{\pi x}{2} \right)}}\right)^{2} \csc^{2}{\left(x - \frac{1}{2} \right)}}$$
4
/ 2/pi*x\\ / 2 \
2*pi*|1 + tan |----|| *\1 - tan (-1/2 + x)/
\ \ 4 //
-------------------------------------------------------------------
2 4
/ 2 \ / 2/pi*x\\ / 2/pi*x\\ 2
\1 + tan (-1/2 + x)/*|1 - tan |----|| *|1 - tan |----|| *tan (pi*x)
\ \ 2 // \ \ 4 //
$$\frac{2 \pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right) \left(\tan^{2}{\left(\frac{\pi x}{4} \right)} + 1\right)^{4}}{\left(1 - \tan^{2}{\left(\frac{\pi x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \tan^{2}{\left(\pi x \right)}}$$
2*pi*cos(-1 + 2*x)
------------------
2
sin (pi*x)
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\sin^{2}{\left(\pi x \right)}}$$
4*pi*cos(-1 + 2*x) ------------------ 1 - cos(2*pi*x)
$$\frac{4 \pi \cos{\left(2 x - 1 \right)}}{1 - \cos{\left(2 \pi x \right)}}$$
2 2/ 1 x\ 4/ 1 x\ 4/pi*x\ / 2 \
pi*cot (pi*x)*cot |- - + -|*sin |- - + -|*tan |----|*\-1 + cot (-1/2 + x)/
\ 4 2/ \ 4 2/ \ 4 /
--------------------------------------------------------------------------
2
/ 2/pi*x\\ 8/pi*x\
2*|-1 + cot |----|| *sin |----|
\ \ 2 // \ 4 /
$$\frac{\pi \left(\cot^{2}{\left(x - \frac{1}{2} \right)} - 1\right) \sin^{4}{\left(\frac{x}{2} - \frac{1}{4} \right)} \tan^{4}{\left(\frac{\pi x}{4} \right)} \cot^{2}{\left(\pi x \right)} \cot^{2}{\left(\frac{x}{2} - \frac{1}{4} \right)}}{2 \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)^{2} \sin^{8}{\left(\frac{\pi x}{4} \right)}}$$
/ 2 \
4/pi*x\ 2/ pi \ | sec (-1/2 + x) |
2*pi*sec |----|*sec |- -- + pi*x|*|1 - ------------------|
\ 2 / \ 2 / | 2/ 1 pi\|
| sec |- - + x - --||
\ \ 2 2 //
------------------------------------------------------------
2
/ 2/pi*x\ \
/ 2 \ | sec |----| |
| sec (-1/2 + x) | | \ 2 / | 2
|1 + ------------------|*|1 - -----------------| *sec (pi*x)
| 2/ 1 pi\| | 2/ pi pi*x\|
| sec |- - + x - --|| | sec |- -- + ----||
\ \ 2 2 // \ \ 2 2 //
$$\frac{2 \pi \left(- \frac{\sec^{2}{\left(x - \frac{1}{2} \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} - \frac{1}{2} \right)}} + 1\right) \sec^{4}{\left(\frac{\pi x}{2} \right)} \sec^{2}{\left(\pi x - \frac{\pi}{2} \right)}}{\left(- \frac{\sec^{2}{\left(\frac{\pi x}{2} \right)}}{\sec^{2}{\left(\frac{\pi x}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \left(\frac{\sec^{2}{\left(x - \frac{1}{2} \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} - \frac{1}{2} \right)}} + 1\right) \sec^{2}{\left(\pi x \right)}}$$
/ pi \
2*pi*sin|-1 + -- + 2*x|
\ 2 /
-----------------------
2
sin (pi*x)
$$\frac{2 \pi \sin{\left(2 x - 1 + \frac{\pi}{2} \right)}}{\sin^{2}{\left(\pi x \right)}}$$
/ 2 \ 2*pi*\1 - tan (-1/2 + x)/ ------------------------------- / 2 \ 2 \1 + tan (-1/2 + x)/*sin (pi*x)
$$\frac{2 \pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right)}{\left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \sin^{2}{\left(\pi x \right)}}$$
2
/ 2/pi*x\\ 2 / 2 \
2*pi*|1 + cot |----|| *cot (pi*x)*\-1 + cot (-1/2 + x)/
\ \ 2 //
-------------------------------------------------------
2
/ 2 \ / 2/pi*x\\
\1 + cot (-1/2 + x)/*|-1 + cot |----||
\ \ 2 //
$$\frac{2 \pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(x - \frac{1}{2} \right)} - 1\right) \cot^{2}{\left(\pi x \right)}}{\left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(x - \frac{1}{2} \right)} + 1\right)}$$
2 / pi \
pi*sin (2*pi*x)*sin|-1 + -- + 2*x|
\ 2 /
----------------------------------
4 2/pi \
2*sin (pi*x)*sin |-- + pi*x|
\2 /
$$\frac{\pi \sin^{2}{\left(2 \pi x \right)} \sin{\left(2 x - 1 + \frac{\pi}{2} \right)}}{2 \sin^{4}{\left(\pi x \right)} \sin^{2}{\left(\pi x + \frac{\pi}{2} \right)}}$$
2
/ 2/pi*x\\ / 2 \
pi*|1 + tan |----|| *\1 - tan (-1/2 + x)/
\ \ 2 //
-----------------------------------------
/ 2 \ 2/pi*x\
2*\1 + tan (-1/2 + x)/*tan |----|
\ 2 /
$$\frac{\pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)^{2}}{2 \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \tan^{2}{\left(\frac{\pi x}{2} \right)}}$$
2 2 / 1 \
pi*(-1 + cos(pi*x)) *sin (-1/2 + x)*|-1 + --------------|
| 2 |
\ tan (-1/2 + x)/
---------------------------------------------------------
2 4/pi*x\
2*sin (pi*x)*sin |----|
\ 2 /
$$\frac{\pi \left(-1 + \frac{1}{\tan^{2}{\left(x - \frac{1}{2} \right)}}\right) \left(\cos{\left(\pi x \right)} - 1\right)^{2} \sin^{2}{\left(x - \frac{1}{2} \right)}}{2 \sin^{4}{\left(\frac{\pi x}{2} \right)} \sin^{2}{\left(\pi x \right)}}$$
/ 2/ 1 pi\\
| cos |- - + x - --||
2 | \ 2 2 /|
2*pi*cos (pi*x)*|1 - ------------------|
| 2 |
\ cos (-1/2 + x) /
------------------------------------------------------------------------------
2
/ 2/ 1 pi\\ / 2/ pi pi*x\\
| cos |- - + x - --|| | cos |- -- + ----||
| \ 2 2 /| | \ 2 2 /| 4/pi*x\ 2/ pi \
|1 + ------------------|*|1 - -----------------| *cos |----|*cos |- -- + pi*x|
| 2 | | 2/pi*x\ | \ 2 / \ 2 /
\ cos (-1/2 + x) / | cos |----| |
\ \ 2 / /
$$\frac{2 \pi \left(1 - \frac{\cos^{2}{\left(x - \frac{\pi}{2} - \frac{1}{2} \right)}}{\cos^{2}{\left(x - \frac{1}{2} \right)}}\right) \cos^{2}{\left(\pi x \right)}}{\left(1 - \frac{\cos^{2}{\left(\frac{\pi x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{\pi x}{2} \right)}}\right)^{2} \left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} - \frac{1}{2} \right)}}{\cos^{2}{\left(x - \frac{1}{2} \right)}}\right) \cos^{4}{\left(\frac{\pi x}{2} \right)} \cos^{2}{\left(\pi x - \frac{\pi}{2} \right)}}$$
/ 2/1 pi \\
| csc |- + -- - x||
2 4/pi pi*x\ | \2 2 /|
2*pi*csc (pi*x)*csc |-- - ----|*|1 - ----------------|
\2 2 / | 2 |
\ csc (-1/2 + x) /
-------------------------------------------------------------
2
/ 2/1 pi \\ / 2/pi pi*x\\
| csc |- + -- - x|| | csc |-- - ----||
| \2 2 /| | \2 2 /| 2/pi \
|1 + ----------------|*|1 - ---------------| *csc |-- - pi*x|
| 2 | | 2/pi*x\ | \2 /
\ csc (-1/2 + x) / | csc |----| |
\ \ 2 / /
$$\frac{2 \pi \left(1 - \frac{\csc^{2}{\left(- x + \frac{1}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x - \frac{1}{2} \right)}}\right) \csc^{2}{\left(\pi x \right)} \csc^{4}{\left(- \frac{\pi x}{2} + \frac{\pi}{2} \right)}}{\left(1 - \frac{\csc^{2}{\left(- \frac{\pi x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{\pi x}{2} \right)}}\right)^{2} \left(1 + \frac{\csc^{2}{\left(- x + \frac{1}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x - \frac{1}{2} \right)}}\right) \csc^{2}{\left(- \pi x + \frac{\pi}{2} \right)}}$$
2 2*pi*csc (pi*x)*cos(-1 + 2*x)
$$2 \pi \cos{\left(2 x - 1 \right)} \csc^{2}{\left(\pi x \right)}$$
2*pi*cos(2*x - 1)
-----------------
2
sin (pi*x)
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\sin^{2}{\left(\pi x \right)}}$$
4*pi ---------------------------------------- / 1 \ / pi \ |1 - ----------------|*csc|1 + -- - 2*x| | /pi \| \ 2 / | csc|-- - 2*pi*x|| \ \2 //
$$\frac{4 \pi}{\left(1 - \frac{1}{\csc{\left(- 2 \pi x + \frac{\pi}{2} \right)}}\right) \csc{\left(- 2 x + 1 + \frac{\pi}{2} \right)}}$$
/ 2 \
2 4/pi*x\ | csc (-1/2 + x) |
2*pi*csc (pi*x)*csc |----|*|-1 + ----------------|
\ 2 / | 2/1 pi \|
| csc |- + -- - x||
\ \2 2 //
------------------------------------------------------
2
/ 2/pi*x\ \
| csc |----| |
| \ 2 / | 2 2/pi \
|-1 + ---------------| *csc (-1/2 + x)*csc |-- - pi*x|
| 2/pi pi*x\| \2 /
| csc |-- - ----||
\ \2 2 //
$$\frac{2 \pi \left(\frac{\csc^{2}{\left(x - \frac{1}{2} \right)}}{\csc^{2}{\left(- x + \frac{1}{2} + \frac{\pi}{2} \right)}} - 1\right) \csc^{4}{\left(\frac{\pi x}{2} \right)} \csc^{2}{\left(\pi x \right)}}{\left(\frac{\csc^{2}{\left(\frac{\pi x}{2} \right)}}{\csc^{2}{\left(- \frac{\pi x}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2} \csc^{2}{\left(x - \frac{1}{2} \right)} \csc^{2}{\left(- \pi x + \frac{\pi}{2} \right)}}$$
2/ pi \
2*pi*sec |- -- + pi*x|
\ 2 /
----------------------
sec(-1 + 2*x)
$$\frac{2 \pi \sec^{2}{\left(\pi x - \frac{\pi}{2} \right)}}{\sec{\left(2 x - 1 \right)}}$$
2 / 2 \
pi*(1 + cos(pi*x)) *\1 - tan (-1/2 + x)/
--------------------------------------------
/ 2 \ 4/pi*x\ 2
2*\1 + tan (-1/2 + x)/*cos |----|*sin (pi*x)
\ 2 /
$$\frac{\pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right) \left(\cos{\left(\pi x \right)} + 1\right)^{2}}{2 \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \sin^{2}{\left(\pi x \right)} \cos^{4}{\left(\frac{\pi x}{2} \right)}}$$
4
/ 2/pi*x\\ 2/ 1 x\ / 1 \
pi*|1 + tan |----|| *tan |- - + -|*|-1 + --------------|
\ \ 4 // \ 4 2/ | 2 |
\ tan (-1/2 + x)/
---------------------------------------------------------------
2 2
/ 2/ 1 x\\ / 1 \ 2 4/pi*x\
2*|1 + tan |- - + -|| *|-1 + ----------| *tan (pi*x)*tan |----|
\ \ 4 2// | 2/pi*x\| \ 4 /
| tan |----||
\ \ 2 //
$$\frac{\pi \left(-1 + \frac{1}{\tan^{2}{\left(x - \frac{1}{2} \right)}}\right) \left(\tan^{2}{\left(\frac{\pi x}{4} \right)} + 1\right)^{4} \tan^{2}{\left(\frac{x}{2} - \frac{1}{4} \right)}}{2 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{\pi x}{2} \right)}}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} - \frac{1}{4} \right)} + 1\right)^{2} \tan^{4}{\left(\frac{\pi x}{4} \right)} \tan^{2}{\left(\pi x \right)}}$$
2 4/pi*x\ / 2 \
2*pi*cos (pi*x)*sec |----|*\1 - tan (-1/2 + x)/
\ 2 /
-------------------------------------------------
2
/ 2 \ / 2/pi*x\\ 2
\1 + tan (-1/2 + x)/*|1 - tan |----|| *sin (pi*x)
\ \ 2 //
$$\frac{2 \pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right) \cos^{2}{\left(\pi x \right)} \sec^{4}{\left(\frac{\pi x}{2} \right)}}{\left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \sin^{2}{\left(\pi x \right)}}$$
/ 2/ 1 pi\\
| sec |- - + x - --||
2/ pi \ 4/ pi pi*x\ | \ 2 2 /|
2*pi*sec |- -- + pi*x|*sec |- -- + ----|*|-1 + ------------------|
\ 2 / \ 2 2 / | 2 |
\ sec (-1/2 + x) /
------------------------------------------------------------------
2
/ 2/ pi pi*x\\
| sec |- -- + ----||
| \ 2 2 /| 2 2/ 1 pi\
|-1 + -----------------| *sec (pi*x)*sec |- - + x - --|
| 2/pi*x\ | \ 2 2 /
| sec |----| |
\ \ 2 / /
$$\frac{2 \pi \left(-1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} - \frac{1}{2} \right)}}{\sec^{2}{\left(x - \frac{1}{2} \right)}}\right) \sec^{4}{\left(\frac{\pi x}{2} - \frac{\pi}{2} \right)} \sec^{2}{\left(\pi x - \frac{\pi}{2} \right)}}{\left(-1 + \frac{\sec^{2}{\left(\frac{\pi x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{\pi x}{2} \right)}}\right)^{2} \sec^{2}{\left(\pi x \right)} \sec^{2}{\left(x - \frac{\pi}{2} - \frac{1}{2} \right)}}$$
2 2*pi*csc (pi*x) ----------------- / pi \ csc|1 + -- - 2*x| \ 2 /
$$\frac{2 \pi \csc^{2}{\left(\pi x \right)}}{\csc{\left(- 2 x + 1 + \frac{\pi}{2} \right)}}$$
2/pi \
2*pi*sec |-- - pi*x|
\2 /
--------------------
sec(-1 + 2*x)
$$\frac{2 \pi \sec^{2}{\left(- \pi x + \frac{\pi}{2} \right)}}{\sec{\left(2 x - 1 \right)}}$$
/ 2 \
4*pi*\-1 + cot (-1/2 + x)/
------------------------------------------
/ 2 \
/ 2 \ | -1 + cot (pi*x)|
\1 + cot (-1/2 + x)/*|1 - ---------------|
| 2 |
\ 1 + cot (pi*x)/
$$\frac{4 \pi \left(\cot^{2}{\left(x - \frac{1}{2} \right)} - 1\right)}{\left(- \frac{\cot^{2}{\left(\pi x \right)} - 1}{\cot^{2}{\left(\pi x \right)} + 1} + 1\right) \left(\cot^{2}{\left(x - \frac{1}{2} \right)} + 1\right)}$$
4*pi ------------------------------- / 1 \ |1 - -----------|*sec(-1 + 2*x) \ sec(2*pi*x)/
$$\frac{4 \pi}{\left(1 - \frac{1}{\sec{\left(2 \pi x \right)}}\right) \sec{\left(2 x - 1 \right)}}$$
4
/ 2/pi*x\\ 2 2/ 1 x\ / 2 \
pi*|1 + cot |----|| *cot (pi*x)*cot |- - + -|*\-1 + cot (-1/2 + x)/
\ \ 4 // \ 4 2/
-------------------------------------------------------------------
2 2
/ 2/ 1 x\\ / 2/pi*x\\ 4/pi*x\
2*|1 + cot |- - + -|| *|-1 + cot |----|| *cot |----|
\ \ 4 2// \ \ 2 // \ 4 /
$$\frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{4} \right)} + 1\right)^{4} \left(\cot^{2}{\left(x - \frac{1}{2} \right)} - 1\right) \cot^{2}{\left(\pi x \right)} \cot^{2}{\left(\frac{x}{2} - \frac{1}{4} \right)}}{2 \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)^{2} \left(\cot^{2}{\left(\frac{x}{2} - \frac{1}{4} \right)} + 1\right)^{2} \cot^{4}{\left(\frac{\pi x}{4} \right)}}$$
2 2 / 2 \
2*pi*cot (pi*x)*sin (-1/2 + x)*\-1 + cot (-1/2 + x)/
----------------------------------------------------
2
/ 2/pi*x\\ 4/pi*x\
|-1 + cot |----|| *sin |----|
\ \ 2 // \ 2 /
$$\frac{2 \pi \left(\cot^{2}{\left(x - \frac{1}{2} \right)} - 1\right) \sin^{2}{\left(x - \frac{1}{2} \right)} \cot^{2}{\left(\pi x \right)}}{\left(\cot^{2}{\left(\frac{\pi x}{2} \right)} - 1\right)^{2} \sin^{4}{\left(\frac{\pi x}{2} \right)}}$$
4
/ 2/pi*x\\ 2 / 1 \
2*pi*|1 + cot |----|| *cot (pi*x)*|1 - --------------|
\ \ 4 // | 2 |
\ cot (-1/2 + x)/
---------------------------------------------------------
2 4
/ 1 \ / 1 \ / 2/pi*x\\
|1 + --------------|*|1 - ----------| *|-1 + cot |----||
| 2 | | 2/pi*x\| \ \ 4 //
\ cot (-1/2 + x)/ | cot |----||
\ \ 2 //
$$\frac{2 \pi \left(1 - \frac{1}{\cot^{2}{\left(x - \frac{1}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{\pi x}{4} \right)} + 1\right)^{4} \cot^{2}{\left(\pi x \right)}}{\left(1 - \frac{1}{\cot^{2}{\left(\frac{\pi x}{2} \right)}}\right)^{2} \left(1 + \frac{1}{\cot^{2}{\left(x - \frac{1}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{\pi x}{4} \right)} - 1\right)^{4}}$$
2 / 2 \
2*pi*cot (pi*x)*\1 - tan (-1/2 + x)/
--------------------------------------------------------------------
2 4
/ 2 \ / 2/pi*x\\ / 2/pi*x\\ 8/pi*x\
\1 + tan (-1/2 + x)/*|1 - tan |----|| *|-1 + cot |----|| *sin |----|
\ \ 2 // \ \ 4 // \ 4 /
$$\frac{2 \pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right) \cot^{2}{\left(\pi x \right)}}{\left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{\pi x}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{\pi x}{4} \right)}}$$
/ 2 \
2*pi*\1 - tan (-1/2 + x)/
------------------------------------------------------------------------------
2 4
/ 2 \ / 2/pi*x\\ / 2/pi*x\\ 8/pi*x\ 2
\1 + tan (-1/2 + x)/*|1 - tan |----|| *|1 - tan |----|| *cos |----|*tan (pi*x)
\ \ 2 // \ \ 4 // \ 4 /
$$\frac{2 \pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right)}{\left(1 - \tan^{2}{\left(\frac{\pi x}{4} \right)}\right)^{4} \left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \cos^{8}{\left(\frac{\pi x}{4} \right)} \tan^{2}{\left(\pi x \right)}}$$
2 2*pi*csc (pi*x) --------------- sec(-1 + 2*x)
$$\frac{2 \pi \csc^{2}{\left(\pi x \right)}}{\sec{\left(2 x - 1 \right)}}$$
2
pi*sin (2*pi*x)*cos(-1 + 2*x)
-----------------------------
2 4
2*cos (pi*x)*sin (pi*x)
$$\frac{\pi \sin^{2}{\left(2 \pi x \right)} \cos{\left(2 x - 1 \right)}}{2 \sin^{4}{\left(\pi x \right)} \cos^{2}{\left(\pi x \right)}}$$
2
/ 2/pi*x\\ / 2 \
pi*|1 + cot |----|| *\-1 + cot (-1/2 + x)/
\ \ 2 //
------------------------------------------
/ 2 \ 2/pi*x\
2*\1 + cot (-1/2 + x)/*cot |----|
\ 2 /
$$\frac{\pi \left(\cot^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(x - \frac{1}{2} \right)} - 1\right)}{2 \left(\cot^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \cot^{2}{\left(\frac{\pi x}{2} \right)}}$$
/ 2 \
4*pi*\1 - tan (-1/2 + x)/
-----------------------------------------
/ 2 \
/ 2 \ | 1 - tan (pi*x)|
\1 + tan (-1/2 + x)/*|1 - --------------|
| 2 |
\ 1 + tan (pi*x)/
$$\frac{4 \pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right)}{\left(- \frac{1 - \tan^{2}{\left(\pi x \right)}}{\tan^{2}{\left(\pi x \right)} + 1} + 1\right) \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right)}$$
/ 2 \
2 2/ 1 pi\ | cos (-1/2 + x) |
2*pi*cos (pi*x)*cos |- - + x - --|*|-1 + ------------------|
\ 2 2 / | 2/ 1 pi\|
| cos |- - + x - --||
\ \ 2 2 //
-------------------------------------------------------------
2
/ 2/pi*x\ \
| cos |----| |
| \ 2 / | 2/ pi \ 4/ pi pi*x\
|-1 + -----------------| *cos |- -- + pi*x|*cos |- -- + ----|
| 2/ pi pi*x\| \ 2 / \ 2 2 /
| cos |- -- + ----||
\ \ 2 2 //
$$\frac{2 \pi \left(\frac{\cos^{2}{\left(x - \frac{1}{2} \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} - \frac{1}{2} \right)}} - 1\right) \cos^{2}{\left(\pi x \right)} \cos^{2}{\left(x - \frac{\pi}{2} - \frac{1}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{\pi x}{2} \right)}}{\cos^{2}{\left(\frac{\pi x}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2} \cos^{4}{\left(\frac{\pi x}{2} - \frac{\pi}{2} \right)} \cos^{2}{\left(\pi x - \frac{\pi}{2} \right)}}$$
2*pi*cos(-1 + 2*x)
------------------
2/ pi \
cos |- -- + pi*x|
\ 2 /
$$\frac{2 \pi \cos{\left(2 x - 1 \right)}}{\cos^{2}{\left(\pi x - \frac{\pi}{2} \right)}}$$
/ 2 \
2 2 | sin (-1 + 2*x) |
pi*sin (2*pi*x)*sin (-1/2 + x)*|-1 + ----------------|
| 4 |
\ 4*sin (-1/2 + x)/
------------------------------------------------------
2
/ 2 \
| sin (pi*x) | 4 4/pi*x\
2*|-1 + ------------| *sin (pi*x)*sin |----|
| 4/pi*x\| \ 2 /
| 4*sin |----||
\ \ 2 //
$$\frac{\pi \left(-1 + \frac{\sin^{2}{\left(2 x - 1 \right)}}{4 \sin^{4}{\left(x - \frac{1}{2} \right)}}\right) \sin^{2}{\left(2 \pi x \right)} \sin^{2}{\left(x - \frac{1}{2} \right)}}{2 \left(-1 + \frac{\sin^{2}{\left(\pi x \right)}}{4 \sin^{4}{\left(\frac{\pi x}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{\pi x}{2} \right)} \sin^{4}{\left(\pi x \right)}}$$
2
/ 2/pi*x\\ / 2 \
2*pi*|1 + tan |----|| *\1 - tan (-1/2 + x)/
\ \ 2 //
-------------------------------------------------
2
/ 2 \ / 2/pi*x\\ 2
\1 + tan (-1/2 + x)/*|1 - tan |----|| *tan (pi*x)
\ \ 2 //
$$\frac{2 \pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right) \left(\tan^{2}{\left(\frac{\pi x}{2} \right)} + 1\right)^{2}}{\left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \tan^{2}{\left(\pi x \right)}}$$
2/ 1 x\ / 1 \
pi*tan |- - + -|*|-1 + --------------|
\ 4 2/ | 2 |
\ tan (-1/2 + x)/
--------------------------------------------------------------------------
2 2
/ 2/ 1 x\\ / 1 \ 8/pi*x\ 2 4/pi*x\
2*|1 + tan |- - + -|| *|-1 + ----------| *cos |----|*tan (pi*x)*tan |----|
\ \ 4 2// | 2/pi*x\| \ 4 / \ 4 /
| tan |----||
\ \ 2 //
$$\frac{\pi \left(-1 + \frac{1}{\tan^{2}{\left(x - \frac{1}{2} \right)}}\right) \tan^{2}{\left(\frac{x}{2} - \frac{1}{4} \right)}}{2 \left(-1 + \frac{1}{\tan^{2}{\left(\frac{\pi x}{2} \right)}}\right)^{2} \left(\tan^{2}{\left(\frac{x}{2} - \frac{1}{4} \right)} + 1\right)^{2} \cos^{8}{\left(\frac{\pi x}{4} \right)} \tan^{4}{\left(\frac{\pi x}{4} \right)} \tan^{2}{\left(\pi x \right)}}$$
/ 2 \
2*pi*\1 - tan (-1/2 + x)/
------------------------------------------------------------
2
/ 2 \ / 2/pi*x\\ 4/pi*x\ 2
\1 + tan (-1/2 + x)/*|1 - tan |----|| *cos |----|*tan (pi*x)
\ \ 2 // \ 2 /
$$\frac{2 \pi \left(1 - \tan^{2}{\left(x - \frac{1}{2} \right)}\right)}{\left(1 - \tan^{2}{\left(\frac{\pi x}{2} \right)}\right)^{2} \left(\tan^{2}{\left(x - \frac{1}{2} \right)} + 1\right) \cos^{4}{\left(\frac{\pi x}{2} \right)} \tan^{2}{\left(\pi x \right)}}$$
2*pi*(1 - tan(-1/2 + x)^2)/((1 + tan(-1/2 + x)^2)*(1 - tan(pi*x/2)^2)^2*cos(pi*x/2)^4*tan(pi*x)^2)